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Factorial and reduced K-means reconsidered

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  • Timmerman, Marieke E.
  • Ceulemans, Eva
  • Kiers, Henk A.L.
  • Vichi, Maurizio

Abstract

Factorial K-means analysis (FKM) and Reduced K-means analysis (RKM) are clustering methods that aim at simultaneously achieving a clustering of the objects and a dimension reduction of the variables. Because a comprehensive comparison between FKM and RKM is lacking in the literature so far, a theoretical and simulation-based comparison between FKM and RKM is provided. It is shown theoretically how FKM's versus RKM's performances are affected by the presence of residuals within the clustering subspace and/or within its orthocomplement in the observed data. The simulation study confirmed that for both FKM and RKM, the cluster membership recovery generally deteriorates with increasing amount of overlap between clusters. Furthermore, the conjectures were confirmed that for FKM the subspace recovery deteriorates with increasing relative sizes of subspace residuals compared to the complement residuals, and that the reverse holds for RKM. As such, FKM and RKM complement each other. When the majority of the variables reflect the clustering structure, and/or standardized variables are being analyzed, RKM can be expected to perform reasonably well. However, because both RKM and FKM may suffer from subspace and membership recovery problems, it is essential to critically evaluate their solutions on the basis of the content of the clustering problem at hand.

Suggested Citation

  • Timmerman, Marieke E. & Ceulemans, Eva & Kiers, Henk A.L. & Vichi, Maurizio, 2010. "Factorial and reduced K-means reconsidered," Computational Statistics & Data Analysis, Elsevier, vol. 54(7), pages 1858-1871, July.
    • Handle: RePEc:eee:csdana:v:54:y:2010:i:7:p:1858-1871
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    3. Masaki Mitsuhiro & Hiroshi Yadohisa, 2015. "Reduced $$k$$ k -means clustering with MCA in a low-dimensional space," Computational Statistics, Springer, vol. 30(2), pages 463-475, June.
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    5. Roberto Rocci & Stefano Gattone & Maurizio Vichi, 2011. "A New Dimension Reduction Method: Factor Discriminant K-means," Journal of Classification, Springer;The Classification Society, vol. 28(2), pages 210-226, July.
    6. Yoshikazu Terada, 2015. "Strong consistency of factorial $$K$$ K -means clustering," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 67(2), pages 335-357, April.
    7. Michio Yamamoto, 2012. "Clustering of functional data in a low-dimensional subspace," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 6(3), pages 219-247, October.
    8. Kim De Roover & Eva Ceulemans & Marieke Timmerman & John Nezlek & Patrick Onghena, 2013. "Modeling Differences in the Dimensionality of Multiblock Data by Means of Clusterwise Simultaneous Component Analysis," Psychometrika, Springer;The Psychometric Society, vol. 78(4), pages 648-668, October.
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    10. Uno, Kohei & Satomura, Hironori & Adachi, Kohei, 2016. "Fixed factor analysis with clustered factor score constraint," Computational Statistics & Data Analysis, Elsevier, vol. 94(C), pages 265-274.
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    12. Luca Greco & Antonio Lucadamo & Pietro Amenta, 2020. "An Impartial Trimming Approach for Joint Dimension and Sample Reduction," Journal of Classification, Springer;The Classification Society, vol. 37(3), pages 769-788, October.

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